Decoupled mild solutions of path-dependent PDEs and IPDEs represented by BSDEs driven by cadlag martingales. - Archive ouverte HAL Access content directly
Journal Articles Potential Analysis Year : 2020

Decoupled mild solutions of path-dependent PDEs and IPDEs represented by BSDEs driven by cadlag martingales.

Abstract

We focus on a class of path-dependent problems which include path-dependent PDEs and Integro PDEs (in short IPDEs), and their representation via BSDEs driven by a cadlag martingale. For those equations we introduce the notion of {\it decoupled mild solution} for which, under general assumptions, we study existence and uniqueness and its representation via the aforementioned BSDEs. This concept generalizes a similar notion introduced by the authors in recent papers in the framework of classical PDEs and IPDEs. For every initial condition $(s,\eta)$, where $s$ is an initial time and $\eta$ an initial path, the solution of such BSDE produces a couple of processes $(Y^{s,\eta},Z^{s,\eta})$. In the classical (Markovian or not) literature the function $u(s,\eta):= Y^{s,\eta}_s$ constitutes a viscosity type solution of an associated PDE (resp. IPDE); our approach allows not only to identify $u$ as the unique decoupled mild solution, but also to solve quite generally the so called {\it identification problem}, i.e. to also characterize the $(Z^{s,\eta})_{s,\eta}$ processes in term of a deterministic function $v$ associated to the (above decoupled mild) solution $u$.
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Dates and versions

hal-01774823 , version 1 (24-04-2018)
hal-01774823 , version 2 (08-03-2019)

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Adrien Barrasso, Francesco Russo. Decoupled mild solutions of path-dependent PDEs and IPDEs represented by BSDEs driven by cadlag martingales.. Potential Analysis, 2020, 53, pp.449-481. ⟨10.1007/s11118-019-09775-x⟩. ⟨hal-01774823v2⟩
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